Fluctuations of eigenvalues for random Toeplitz and related matrices

Dangzheng Liu,Xin Sun,Zhengdong Wang

doi:10.1214/ejp.v17-2006

Abstract

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,\cdots,$ being independent real random variables such that $$ \mathbb{E}[a_{j}]=0, \mathbb{E} [|a_{j}|^{2}]=1 \mathrm{for}\, j=0,1,2,\cdots,$$ (homogeneity of 4-th moments) $$\kappa=\mathbb{E} [|a_{j}|^{4}],$$ and further (uniform boundedness) $$\sup\limits_{j\geq 0} \mathbb{E} [|a_{j}|^{k}]=C_{k}<\infty \mathrm{for} k\geq 3.$$ Under the assumption of $a_{0}\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed polynomial with degree at least 2. Without this assumption, the CLT can be easily modified to a possibly non-normal limit law. In a special case where $a_{j}$'s are Gaussian, the result has been obtained by Chatterjee for some test functions. Our derivation is based on a simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our method can apply to other related random matrix models, including Hermitian Toeplitz and symmetric Hankel matrices. Since Toeplitz matrices are quite different from Wigner and Wishart matrices, our results enrich this topic.

Highlights

Introduction and main resultsToeplitz matrices appear very often in mathematics and physics and in plenty of applications, see Grenander and Szego’s book [14] for a detailed introduction to deterministic Toeplitz matrices
The study of fluctuations of eigenvalues for random Toeplitz matrices is quite little, to the best of our knowledge, the only known result comes from Chatterjee [9] in the special case where the matrix entries are Gaussian distributions
In the special case that the matrix entries aj are Gaussian distributions, by using his notion of “second order Poincaré inequalities” Chatterjee in [9] proved the following theorem: Theorem ([9], Theorem 4.5) Consider the Gaussian Toeplitz matrices Tn =ni,j=1, i.e. aj = a−j for j = 1, 2, · · ·, and {aj }∞ j=0 is a sequence of independent standard

Summary

Introduction and main results

Toeplitz matrices appear very often in mathematics and physics and in plenty of applications, see Grenander and Szego’s book [14] for a detailed introduction to deterministic Toeplitz matrices. In this paper we will derive a central limit theorem (CLT for short) for linear statistics of eigenvalues of random Toeplitz and related matrices. The basic model under consideration consists of n × n random symmetric Toeplitz band matrices Tn = (ηij ai−j )ni,j=1 in Eq (1.3). In the special case that the matrix entries aj are Gaussian distributions, by using his notion of “second order Poincaré inequalities” Chatterjee in [9] proved the following theorem: Theorem ([9], Theorem 4.5) Consider the Gaussian Toeplitz matrices Tn = (ai−j )ni,j=1, i.e. aj = a−j for j = 1, 2, · · · , and {aj }∞ j=0 is a sequence of independent standard.

Integrals associated with pair partitions

Mathematical expectation

Extensions to other models

Hermitian Toeplitz band matrices

Hankel band matrices